Shear induced drainage of a foamy yield stress fluid is investigated using MRI techniques. Whereas the yield stress of the interstitial fluid stabilizes the system at rest, a fast drainage is observed when a horizontal shear is imposed. It is shown that the sheared interstitial material behaves as a viscous fluid in the direction of gravity, the effective viscosity of which is controlled by shear in transient foam films between bubbles. Results provided for several bubble sizes are not captured by the R^2 scaling classically observed for liquid flow in particulate systems, such as foams and thus constitute a remarkable demonstration of the strong coupling of drainage flow and shear induced interstitial flow. Furthermore, foam films are found to be responsible for the unexpected arrest of drainage, thus trapping irreversibly a significant amount of interstitial liquid.